# Basic category theory by van Oosten J.

By van Oosten J.

Similar algebra & trigonometry books

An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra

During this creation to commutative algebra, the writer choses a path that leads the reader throughout the crucial principles, with out getting embroiled in technicalities. he's taking the reader speedy to the basics of complicated projective geometry, requiring just a easy wisdom of linear and multilinear algebra and a few hassle-free crew conception.

Inequalities : a Mathematical Olympiad approach

This booklet is meant for the Mathematical Olympiad scholars who desire to arrange for the learn of inequalities, a subject matter now of widespread use at quite a few degrees of mathematical competitions. during this quantity we current either vintage inequalities and the extra necessary inequalities for confronting and fixing optimization difficulties.

Recent Progress in Algebra: An International Conference on Recent Progress in Algebra, August 11-15, 1997, Kaist, Taejon, South Korea

This quantity offers the court cases of the foreign convention on ""Recent development in Algebra"" that used to be held on the Korea complicated Institute of technological know-how and expertise (KAIST) and Korea Institute for complicated learn (KIAS). It introduced jointly specialists within the box to debate growth in algebra, combinatorics, algebraic geometry and quantity conception.

Extra info for Basic category theory

Example text

Exercise 69 Suppose f : X → Y is an arrow in a regular category. For m a subobject M of X, represented by a mono E → X, write ∃f (M ) for the subobject Im(f m) of Y . a) Show that ∃f (M ) is well-defined, that is: depends only on M , not on the representative m. b) Show that if M ∈ Sub(X) and N ∈ Sub(Y ), then ∃f (M ) ≤ N if and only if M ≤ f ∗ (N ). 2 The logic of regular categories The fragment of first order logic we are going to interpret in regular categories is the so-called regular fragment.

One of the points of categorical logic is now, that categorical statements about objects and arrows in C can be reformulated as statements about the truth of certain sequents in L(C). You should read the relevant sequents as expressing that we can “do as if the category were Set”. Examples a) C is a terminal object of C if and only if the sequents ∃x(x = x) are valid, where x, y variables of sort C; b) the arrow f : A → B is mono in C if and only if the sequent f (x) = f (y) x,y x = y is true; c) The square f A x,y x = y and /B g h  C  /D d is a pullback square in C if and only if the sequents h(xB ) = d(y C ) x,y ∃z A (f (z) = x ∧ g(z) = y) and f (z A ) = f (z A ) ∧ g(z A ) = g(z A ) z,z z=z are true; d) the fact that f : A → B is epi can not similarly be expressed!

2. In chapter 3 we’ve seen that f is epi iff is a pushout; since left adjoints preserve identities and pushouts, they preserve epis; therefore the forgetful functor Mon → Set does not have a right adjoint; b) The functor (−) × X : Set → Set does not preserve the terminal object unless X is itself terminal in Set; therefore, it does not have a left adjoint for non-terminal X. c) The forgetful functor Pos → Set has a left adjoint, but it cannot have a right adjoint: it preserves all coproducts, including the initial object, but not all coequalizers.